Robustness of an Adaptive Output Feedback for an Anti-Damped Boundary Wave PDE in Presence of In-Domain Viscous Damping

This paper presents a robustness result for the use of a previously developed adaptive output-feedback controller designed for a pure wave PDE. This one-dimensional wave PDE has a boundary actuation opposite to an unknown anti-damping dynamics boundary. We prove that stabilization is still valid in the case of in-domain viscous damping using a controller designed neglecting the damping, provided that the damping coefficient is small enough. Our robustness proof grounds on the use of an alternative pure wave PDE, neglecting the in-domain damping. We compare these two systems and apply a tailored backstepping transformation to carry out Lyapunov analysis. Numerical simulations show the relevance of our result for drilling applications and illustrate the merits of this control design.